We assume that there are two underlying assets available to the insurer and reinsurer: one risk-free asset (a bank account) and one risky asset (a stock). The evolution of the value of the risk-free asset, $S_t\left(t\right)$, over time is given by

$\text{d}{S}_0\left(t\right)=r_0{S}_0\left(t\right)\text{d}t,t\ge 0,$(1)

where $r_0$ represents the risk-free interest rate.

Let the price of the underlying stock (risky asset) at time $t$ be $S_1\left(t\right)$ and suppose that the value of the risky asset at time $t$ is described by the stochastic differential equation(SDE)

$\text{d}{S}_1\left(t\right)=S_1\left(t\right)\left(\mu \text{d}t+\sigma \text{d}{W}_1\left(t\right)\right),t\ge 0,$(2)

where $\mu$ is the appreciation rate of the stock and $\sigma$ is the volatility rate, both $\mu$ and $\sigma$ being positive constants, and $W_1\left(t\right)$ is a standard Brownian motion. To exclude arbitrage opportunities, we assume that $\mu >r_0$.

Consider a plan that includes both active and retired members. Active members contribute to the pension fund, while retired members draw benefits from it. All participants enter the TBP at age $a$ and retire at age $r$, and their survival is described by a function $s\left(x\right)$ with $s\left(a\right)=1$ for $ax$. For ages $a\le x<r$, members may exit the plan due to death or other causes; once $x\ge r$, death is the sole decrement.

Definition 2.1. The legacy insurance model under investigation is formally defined by the following 6-tuple:

$\left\{n\left(t\right),\mathcal{A}\left(t\right),ℛ\left(t\right),\tilde{L}\left(x,t\right),B\left(t\right),C\left(t\right)\right\}$

where

1) Let $n\left(t\right)$ denote the number of new entrants aged a who join the plan at time $t$.

2) The total active membership at time $t$, $\mathcal{A}\left(t\right)$, represents the population of active members currently contributing to the pension fund is

$\mathcal{A}\left(t\right)={\displaystyle {\int }_a^{r}n}\left(t-\left(x-a\right)\right)s\left(x\right)\text{d}x;$

3) The total count of retired members at time $t$ is represented by $ℛ\left(t\right)$,

$ℛ\left(t\right)={\displaystyle {\int }_r^{\omega }n}\left(t-\left(x-a\right)\right)s\left(x\right)\text{d}x;$

4) Let $\tilde{L}\left(x,t\right)$ denote the retirement salary of an individual who is age $x$ at time $t$. It is specified by

$\tilde{L}\left(x,t\right)=L\left(t\right){\text{e}}^{-\alpha \left(x-r\right)},t\ge 0,x\ge r$(3)

where $L\left(t\right)$ represents the annual salary rate for a member retiring at time $t$. The process $L\left(t\right)$ follows $\text{d}L\left(t\right)=L\left(t\right)\left(\alpha \text{d}t+\eta \text{d}{W}_2\left(t\right)\right)$, $t\ge 0$. Here, $\alpha \in {ℝ}^{+}$ is the expected instantaneous growth rate of salary, $\eta \in ℝ$ is the instantaneous volatility, and $W_2$ is a standard Brownian motion. The Brownian motion $W_2$ is correlated with $W_1$ under $\mathbb{P}$ with correlation coefficient.

5) Let $B\left(t\right)$ denote the total benefit payment rate to all retirees at time $t$. It is given by the expression:

$B\left(t\right)={\displaystyle {\int }_r^{\omega }n}\left(t-\left(x-a\right)\right)s\left(x\right)f\left(t\right)L\left(t\right){\text{e}}^{-\left(\alpha -\zeta \right)\left(x-r\right)}\text{d}x$ (4)

where $\zeta$ is the constant annual rate at which the plan grants cost-of-living adjustments to pensions, and $f\left(t\right)$ is a control variable chosen dynamically by the plan trustees.

6) $C\left(t\right)$ denotes the aggregate instantaneous contribution rate for all active members at time $t$, defined as follows:

$C\left(t\right)={\displaystyle {\int }_a^{r}n}\left(t-x+a\right)s\left(x\right)c_0{\text{e}}^{\alpha t}\text{d}x,t\ge 0,$(5)

Remark 2.1.

1) For further technical particulars regarding the model specification and implementation, the reader is directed to consult Wang et al. (2018) in the reference list.

2) For notation convenience, let $C\left(t\right)=C_1\left(t\right){\text{e}}^{\alpha t}$, where $C_1\left(t\right)=c_0{\displaystyle {\int }_a^{r}n}\left(t-\left(x-a\right)\right)s\left(x\right)\text{d}x$. And let $B\left(t\right)=I\left(t\right)f\left(t\right)L\left(t\right)$, where $I\left(t\right)$ is a positive function of $t$, defined as

$I\left(t\right)={\displaystyle {\int }_r^{\omega }n}\left(t-\left(x-a\right)\right)s\left(x\right){\text{e}}^{-\left(\alpha -\zeta \right)\left(x-r\right)}\text{d}x$(6)

In this subsection we derive the wealth dynamics of the pension fund and the reinsurer for the model outlined above, taking into account investment and reinsurance strategies, contributions from active members, and benefit payments to retirees.

Assume that the pension fund can invest in both the risk-free and risky assets described by (1) and (2), respectively, and use the fund to pay retirement benefits. Let $x_0$ denote the initial wealth of this fund, ${\pi }_1\left(t\right)$ denote the amount that the plan manager invests in the risky asset at time $t$, $q\left(t\right)$ denote the fraction of benefits paid by the reinsurance company , $\tilde{c}C\left(t\right)$ denote the fraction of contribution paid by the pension fund to the reinsurance company and Let $X\left(t\right)$ be the pension funds wealth at time $t$ after implementing the investment strategy ${\pi }_1\left(t\right)$ and the reinsurance strategy $q\left(t\right)$. The funds value then evolves according to the following dynamics:

$\begin{array}{l}\{\begin{array}{l}\text{d}X\left(t\right)={\pi }_1\left(t\right)\frac{\text{d}{S}_1\left(t\right)}{S_1\left(t\right)}+\left(X\left(t\right)-{\pi }_1\left(t\right)\right)\frac{\text{d}{S}_0\left(t\right)}{S_0\left(t\right)}+\left[\left(1-\tilde{c}\right)C\left(t\right)-\left(1-q\left(t\right)\right)B\left(t\right)\right]\text{d}t,\\ X\left(0\right)=x_0,\end{array}\hfill \end{array}$ (7)

Using Equations (1), (2), (4) and (5), Equation (7) can be rewritten as follows:

$\begin{array}{l}\{\begin{array}{l}\text{d}X\left(t\right)=\left[r_{0}X\left(t\right)+\left(\mu -r_0\right){\pi }_1\left(t\right)+\left(1-\tilde{c}\right)C_1\left(t\right){\text{e}}^{\alpha \left(t\right)}-\left(1-q\left(t\right)\right)I\left(t\right)f\left(t\right)L\left(t\right)\right]\text{d}t\\ +{\pi }_1\left(t\right)\sigma \text{d}W\left(t\right),\\ X\left(0\right)=x_0,\end{array}\hfill \end{array}$ (8)

Let ${\psi }_1={\left\{\left({\pi }_1\left(s\right),f\left(s\right)\right)\right\}}_{s\in \left[t,T\right]}$ denote the strategy the pension fund follows on the interval $\left[t,T\right]$. Each pair consists of the investment decision ${\pi }_1\left(s\right)$, the amount allocated to the risky asset at time $s$, and the benefit adjustment factor $f\left(s\right)$ applied at that time. Below we give the formal definition of an admissible strategy for the stochastic differential Equation (8).

Definition 2.2. For a fixed $t\in \left[0,T\right]$, a strategy

${\psi }_1={\left\{\left({\pi }_1\left(s\right),f\left(s\right)\right)\right\}}_{s\in \left[t,T\right]}$

is said to be admissible if

i) ${\psi }_1$ is $F_t$-adapted;

ii) $\forall s\in \left[t,T\right],f\left(s\right)\ge 0$ and $E\left[{\displaystyle {\int }_t^T{\left[{\pi }_1\left(s\right)\right]}^2\text{d}s}\right]<+\infty$;

iii) $\left(X^{{\psi }_1},{\psi }_1\right)$ is the unique solution to SDE (8).

Let ${\Psi }_1$ be a set of all admissible strategy ${\psi }_1$.

The target for this continuous-time asset allocation and benefit distribution problem is to minimize expected discounted losses over the remaining time period until time $T$ , where the losses correspond to the benefit risk and the discontinuity risk defined above. Since participants are concerned with both shortfall risk (benefit payments falling below the target) and benefit instability (deviations in either direction from the target), the loss function includes both linear and quadratic terms for benefit risk. Let $J_1\left(t,x,l\right)$ denote the objective function at time $t$, where the fund value and the salary level at time $t$ are $x$ and $l$, respectively. It is defined as follows:

$\begin{array}{l}\{\begin{array}{l}{J}_1\left(t,x,l;{\pi }_1(\cdot ),f(\cdot ),q(\cdot )\right)=E^{{\pi }_1,f,q}\{{\displaystyle {\int }_t^T[{\left(\left(1-q\left(s\right)\right)B\left(s\right)-B^{\text{*}}{\text{e}}^{\beta s}\right)}^2}\\ -{\lambda }_1\left(\left(1-q\left(s\right)\right)B\left(s\right)-B^{\text{*}}{\text{e}}^{\beta s}\right)]{\text{e}}^{-r_{0}s}\text{d}s+{\lambda }_2{\left(X\left(T\right)-x_0{\text{e}}^{r_{0}T}\right)}^2{\text{e}}^{-r_{0}T}\},\\ J_1\left(T,x,l;{\pi }_1(\cdot ),f(\cdot ),q(\cdot )\right)={\lambda }_2{\left(X\left(T\right)-x_0{\text{e}}^{r_{0}T}\right)}^2{\text{e}}^{-r_{0}T},\end{array}\hfill \end{array}$(9)

This mathematical formulation to the practical goals of TBP trustees mentioned in the Introduction. Both ${\lambda }_1$ and ${\lambda }_2$ are nonnegative constants that serve as penalty weights: ${\lambda }_1$ penalizes negative deviations of $B\left(s\right)$ from $B^{\text{*}}{\text{e}}^{\beta s}$, while ${\lambda }_2$ penalizes failure to reach the terminal fund target. The expectations in (9) are conditional on the state variables $x$ and $l$ at time $t$. The values of ${\lambda }_1$ and ${\lambda }_2$ capture the trade-offs chosen by the TBPs stakeholders (for example, the employer, different member groups, and regulators) among benefit stability (weighted by ${\lambda }_1$), benefit adequacy (weighted by ${\lambda }_1$) and intergenerational equity (weighted by ${\lambda }_2$). These penalty weights are a design choice of the plan and cannot be altered by the pension fund.

Assume the reinsurance fund can allocate resources to both a risk-free asset and a risky asset, and that it contributes toward paying some portion of retirement benefits. Let $y_0$ be the fund’s initial capital. At time $t$, let ${\pi }_2\left(t\right)$ be the amount invested in the risky asset and let $Y\left(t\right)$ denote the fund’s wealth under the investment strategy ${\pi }_2\left(t\right)$and the reinsurance strategy $q\left(t\right)$. The evolution of the funds wealth is then governed by the following stochastic dynamics:

$\begin{array}{l}\{\begin{array}{l}\text{d}Y\left(t\right)={\pi }_2\left(t\right)\frac{\text{d}{S}_1\left(t\right)}{S_1\left(t\right)}+\left(Y\left(t\right)-{\pi }_2\left(t\right)\right)\frac{\text{d}{S}_0\left(t\right)}{S_0\left(t\right)}+\left[\tilde{c}C\left(t\right)-q\left(t\right)B\left(t\right)\right]\text{d}t,\\ Y\left(0\right)=y_0,\end{array}\hfill \end{array}$(10)

Using (1), (2), (4) and (5), we can easily rewrite (10) as

$\begin{array}{l}\{\begin{array}{l}\text{d}Y\left(t\right)=\left[r_{0}Y\left(t\right)+\left(\mu -r_0\right){\pi }_2\left(t\right)+\tilde{c}{C}_1\left(t\right){\text{e}}^{\alpha \left(t\right)}-q\left(t\right)I\left(t\right)f\left(t\right)L\left(t\right)\right]\text{d}t\\ +{\pi }_2\left(t\right)\sigma \text{d}W\left(t\right),\\ Y\left(0\right)=y_0,\end{array}\hfill \end{array}$(11)

Let ${\psi }_2={\left\{\left({\pi }_2\left(s\right),q\left(s\right)\right)\right\}}_{s\in \left[t,T\right]}$ denote the strategy employed by the pension fund over the interval $\left[t,T\right]$. Each pair $\left({\pi }_2\left(s\right),q\left(s\right)\right)$ consists of the investment amount ${\pi }_2\left(s\right)$ allocated to the risky asset at time $s$ and the reinsurance parameter $q\left(s\right)$ at time $s$. Below we give the definition of an admissible strategy in relation to the SDE (11).

Definition 2.3. For any fixed $t\in \left[0,T\right]$, a strategy

${\psi }_2={\left\{\left({\pi }_2\left(s\right),q\left(s\right)\right)\right\}}_{s\in \left[t,T\right]}$

is said to be admissible if

i) ${\psi }_2$ is $F_t$-adapted;

ii) $\forall s\in \left[t,T\right],q\left(s\right)\ge 0$ and $E\left[{\displaystyle {\int }_t^T{\left[{\pi }_2\left(s\right)\right]}^2\text{d}s}\right]<+\infty$;

iii) $\left(X^{{\psi }_2},{\psi }_2\right)$ is the unique solution to SDE (11).

Let ${\Psi }_2$ be a set of all admissible strategy ${\psi }_2$.

Similarly, the goal for the reinsurer is to minimize expected discounted losses over the remaining time period until time $T$, where the losses correspond to the discontinuity risk. Let $J_2\left(t,y,l\right)$ denote the objective function evaluated at time $t$, when the pension fund’s wealth is $y$ and the salary level is $l$. It is defined by the expression:

$\begin{array}{l}\{\begin{array}{l}{J}_2\left(t,y,l;{\pi }_2(\cdot ),f(\cdot ),q(\cdot )\right)=E^{{\pi }_2,f,q}\{{\displaystyle {\int }_t^T\left[{\left(\frac{q\left(s\right)}{1-q\left(s\right)}-\theta \left(s\right)\right)}^2+{\lambda }_3\left(\frac{q\left(s\right)}{1-q\left(s\right)}-\theta \left(s\right)\right)\right]\text{d}s}\\ \begin{array}{c}\stackrel{}{}\\ \end{array}+{\lambda }_4{\left(Y\left(T\right)-y_0{\text{e}}^{r_{0}T}\right)}^2{\text{e}}^{-r_{0}T}\},\\ J_2\left(T,y,l;{\pi }_2(\cdot ),f(\cdot ),q(\cdot )\right)={\lambda }_4{\left(Y\left(T\right)-y_0{\text{e}}^{r_{0}T}\right)}^2{\text{e}}^{-r_{0}T},\end{array}\hfill \end{array}$ (12)

where ${\left(\frac{q\left(s\right)}{1-q\left(s\right)}-\theta \left(s\right)\right)}^2$ is a penalty for the effective reinsurance premium rate deviating from a target safety loading. Let $\theta \left(s\right)>0$ denote the reinsurance safety loading at time $s$. The constants ${\lambda }_3$ and ${\lambda }_4$ are nonnegative penalty weights: ${\lambda }_3$ penalizes the nonnegative deviation between $\frac{q\left(s\right)}{1-q\left(s\right)}$ and $\theta \left(s\right)$, while ${\lambda }_4$ penalizes failure to reach the terminal fund target. For notation convenience, let $J_1\left(t,y,l;{\pi }_1(\cdot ),f(\cdot ),q(\cdot )\right)$ and $J_2\left(t,y,l;{\pi }_2(\cdot ),f(\cdot ),q(\cdot )\right)$ be equivalent to $J_1\left(t,y,l;{\psi }_1,{\psi }_2\right)$ and $J_2\left(t,y,l;{\psi }_1,{\psi }_2\right)$.

In this section we apply standard techniques to solve the optimal control problem in (16) and obtain the followers best-response strategy ${\psi }_1^{\text{*}}=\left({\pi }_1^{{\psi }_2}(\cdot ),f^{{\psi }_2}(\cdot )\right)$ among all admissible policies ${\Psi }_1$.

The value function of pension fund is defined as

${\varphi }^1\left(t,x,l\right):=\underset{{\psi }_1\in {\Psi }_1}{\min }{J}_1\left(t,y,l;{\psi }_1,{\psi }_2\right),t,x,l>0$(17)

where $J_1\left(t,x,l\right)$ is given by (9).

First, we derive the Hamilton-Jacobi-Bellman (HJB) equation corresponding to the stochastic control problem (17). For further background see, for example, Fleming and Soner (2006) . Applying variational arguments together with Its formula yields the HJB equation satisfied by the value function ${\varphi }^1\left(t,x,l\right)$:

$\begin{array}{l}\underset{{\psi }_1\in {\Psi }_1}{\min }\{{\varphi }_t^1+\left[r_{0}x+\left(\mu -r_0\right){\pi }_1+\left(1-\tilde{c}\right)C_1\left(t\right){\text{e}}^{\alpha t}-\left(1-q\left(t\right)\right)flI\left(t\right)\right]{\varphi }_x^1{{\displaystyle }}^{}\\ +\alpha l{\varphi }_l^1+\frac{1}{2}{\pi }^2{\sigma }^2{\varphi }_{xx}^1+\frac{1}{2}{\eta }^2{l}^2{\varphi }_{ll}^1+\rho \sigma \eta l\pi {\varphi }_{xl}^1\\ +\left[{\left(\left(1-q\left(t\right)\right)flI\left(t\right)-B^{*}{\text{e}}^{\beta t}\right)}^2-{\lambda }_1\left(\left(1-q\left(t\right)\right)flI\left(t\right)-B^{*}{\text{e}}^{\beta t}\right)\right]{\text{e}}^{-r_{0}t}\}=0\end{array}$ (18)

with the boundary condition

${\varphi }^1\left(T,x,l\right)={\lambda }_2{\left(x-x_0{\text{e}}^{r_{0}T}\right)}^2{\text{e}}^{-r_{0}T},$(19)

where ${\varphi }_t^1,{\varphi }_x^1,{\varphi }_{xx}^1,{\varphi }_l^1,{\varphi }_{ll}^1$ and ${\varphi }_{xl}^1$ are partial derivatives of ${\varphi }^1\left(t,x,l\right)$.

We summarize the solution to the optimal control problem (17) in the following theorem. For notational convenience, denote the Sharpe ratio of the risky asset by $\delta =\left(\mu -r_0\right)/\sigma$.

Theorem 3.1 For the optimal control problem (17), the optimal asset allocation ${\pi }_1^{{\psi }_2}$ and the benefit-adjustment policy $f^{{\psi }_2}$ are given by the following expressions, respectively:

${\pi }_1^{{\psi }_2}\left(t,x,l\right)=-\frac{\delta }{\sigma }\left[x+\frac{Q^1\left(t\right)}{2}\right],$(20)

$f^{{\psi }_2}\left(t,x,l\right)=\frac{1}{l\left(1-q\left(t\right)\right)I\left(t\right)}\left[B^{\text{*}}{\text{e}}^{\beta t}+\frac{{\lambda }_1}{2}+{\lambda }_2{P}^1\left(t\right)\left(x+\frac{Q^1\left(t\right)}{2}\right)\right],$(21)

and the corresponding value function is given by

${\varphi }^1\left(t,x,l\right)={\lambda }_2{\text{e}}^{-r_{0}t}{P}^1\left(t\right)\left[x^2+x{Q}^1\left(t\right)\right]+K^1\left(t\right),$(22)

where

$\begin{array}{l}{K}^1\left(t\right)=x_0^2{\text{e}}^{2r_{0}T}+{\displaystyle {\int }_t^T{\text{e}}^{-r_{0}s}}\{{\lambda }_2{P}^1\left(s\right)Q^1\left(s\right)[\left(1-\tilde{c}\right)C_1\left(s\right){\text{e}}^{\alpha s}\stackrel{}{}\\ -B^{\text{*}}{\text{e}}^{\beta s}-\frac{{\lambda }_1}{2}-\frac{1}{4}\left({\delta }^2+{\lambda }_2{P}^1\left(s\right)\right)Q^1\left(s\right)]-\frac{{\lambda }_1^2}{4}\}\text{d}s,\end{array}$(23)

and expressions of $P^1\left(t\right)$ and $Q^1\left(t\right)$ are given below, depending on the values of the parameters $\beta$, $r_0$ and ${\sigma }^2$,

$P^1\left(t\right)=\begin{array}{l}\{\begin{array}{ll}\frac{1}{{\lambda }_2\left(T-t\right)+1},\hfill & r_0={\delta }^2,\hfill \\ \frac{r_0-{\delta }^2}{{\lambda }_2+\left(r_0-{\delta }^2-{\lambda }_2\right){\text{e}}^{-\left(r-{\sigma }^2\right)\left(T-t\right)}},\hfill & r_0\ne {\delta }^2,\hfill \end{array}\hfill \end{array}$(24)

$Q^1\left(t\right)=\begin{array}{l}\{\begin{array}{ll}2{\text{e}}^{r_{0}t}\left[{\displaystyle {\int }_t^T\left(1-\tilde{c}\right)C_1\left(s\right){\text{e}}^{\left(\alpha -r_0\right)s}\text{d}s}-B^{*}\left(T-t\right)-x_0\right]\hfill & \hfill \\ +\frac{{\lambda }_1}{r_0}\left({\text{e}}^{-r_0\left(T-t\right)}-1\right),\hfill & r_0=\beta ,\hfill \\ 2{\text{e}}^{r_{0}t}[{\displaystyle {\int }_t^T\left(1-\tilde{c}\right)C_1\left(s\right){\text{e}}^{\left(\alpha -r_0\right)s}\text{d}s}-x_0\begin{array}{c}\\ \end{array}\hfill & \hfill \\ -\frac{B^{*}}{\beta -r_0}\left({\text{e}}^{\left(\beta -r_0\right)T}-{\text{e}}^{\left(\beta -r_0\right)t}\right)]+\frac{{\lambda }_1}{r_0}\left({\text{e}}^{-r_0\left(T-t\right)}-1\right),\hfill & r_0\ne \beta ,\hfill \end{array}\hfill \end{array}$(25)

Proof. First, observe that the minimization in (18) with respect to ${\pi }_1$ and $f$ can be separated into two coupled subproblems: one minimizing with respect to ${\pi }_1$ and the other with respect to $f$. These two dependent minimization problems are written as follows.

$\begin{array}{l}\underset{{\pi }_1\in {\Pi }_1}{\min }\{{\varphi }_t^1+\left[r_{0}x+\left(\mu -r_0\right){\pi }_1+\left(1-\tilde{c}\right)C_1\left(t\right)e^{\alpha t}\right]{\varphi }_x^1+\alpha l{\varphi }_l^1\stackrel{}{}\\ +\frac{1}{2}{\pi }^2{\sigma }^2{\varphi }_{xx}^1+\rho \sigma \eta l\pi {\varphi }_{xl}^1\}=0,\end{array}$(26)

$\begin{array}{l}\underset{f\in F}{\text{min}}\{-\left(1-q\left(t\right)\right)flI\left(t\right){\varphi }_x^1+\frac{1}{2}{\eta }^2{l}^2{\varphi }_l^{1}l+[{\left(\left(1-q\left(t\right)\right)flI\left(t\right)-B^{\text{*}}{\text{e}}^{\beta t}\right)}^2\\ \underset{}{}-{\lambda }_1\left(\left(1-q\left(t\right)\right)flI\left(t\right)-B^{\text{*}}{\text{e}}^{\beta t}\right)]{\text{e}}^{-r_{0}t}\}=0,\end{array}$(27)

Differentiating the bracketed expression in (26) with respect to ${\pi }_1$, and the bracketed expression in (27) with respect to $f$, then setting the derivatives to zero and solving, yields immediately:

${\pi }_1^{{\psi }_2}\left(t,x,l\right)=\frac{\delta {\varphi }_x^1+\rho \eta l{\varphi }_{xl}^1}{\sigma {\varphi }_{xx}^1},$(28)

$f^{{\psi }_2}\left(t,x,l\right)=\frac{1}{l\left(1-q\left(t\right)\right)I\left(t\right)}\left[\frac{{\varphi }_x{\text{e}}^{r_{0}t}+{\lambda }_1}{2}+B^{\text{*}}{\text{e}}^{\beta t}\right],$(29)

Obviously, a sufficient condition for ${\pi }_1$ to be minimal is:

${\varphi }_{xx}^1>0.$(30)

We will confirm this condition after deriving the expression for ${\varphi }^1$.

Next we derive an explicit formula for ${\varphi }^1\left(t,x,l\right)$. Guided by the terminal condition (19), we assume ${\varphi }^1\left(t,x,l\right)$ takes the form:

$\begin{array}{c}{\varphi }^1\left(t,x,l\right)={\lambda }_2{\text{e}}^{-r_{0}t}{P}^1\left(t\right)\left[x^2+Q^1\left(t\right)x\right]+R^1\left(t\right)xl+U^1\left(t\right)l^2\\ +V^1\left(t\right)l+K^1\left(t\right),\end{array}$(31)

where $P^1\left(t\right)$, $Q^1\left(t\right)$, $R^1\left(t\right)$, $U^1\left(t\right)$, $V^1\left(t\right)$, $K^1\left(t\right)$ are functions of t to be determined. The boundary condition (19) implies that

$\begin{array}{l}{P}^1\left(T\right)=1,Q^1\left(T\right)=-2x_0{\text{e}}^{r_{0}T},K^1\left(T\right)=x_0^2{\text{e}}^{2r_{0}T},\\ R^1\left(T\right)=U^1\left(T\right)=V^1\left(T\right)=0,\end{array}$(32)

From (31) we obtain

$\begin{array}{l}{\varphi }_t^1={\lambda }_2{\text{e}}^{-r_{0}t}\left\{-r_0{P}^1\left(t\right)\left[x^2+Q^1\left(t\right)x\right]+P^1\left(t\right)Q_t^{1}x+P_t^1\left[x^2+Q^1\left(t\right)x\right]\right\}\\ +R_t^{1}xl+U_t^1{l}^2+V_t^{1}l+K_t^1,\\ {\varphi }_x^1={\lambda }_2{\text{e}}^{-r_{0}t}{P}^1\left(t\right)\left[2x+Q^1\left(t\right)\right]+R^1\left(t\right)l\\ {\varphi }_l^1=R^1\left(t\right)x+2U^1\left(t\right)l+V^1\left(t\right)\\ {\varphi }_{xx}^1=2{\lambda }_2{\text{e}}^{-r_{0}t}{P}^1\left(t\right),{\varphi }_{ll}^1=2U^1\left(t\right),{\varphi }_{xl}^1=R^1\left(t\right).\end{array}$(33)

Substituting (33) into (28) and (29), the optimal controls can be written in terms of the time-dependent coefficients $P^1\left(t\right),Q^1\left(t\right)$ and $R^1\left(t\right)$. Inserting (33), together with (28) and (29), into the HJB equation (18) and collecting coefficients of the powers and cross-terms $x^2$, $l^2$, $x^l$, $x$, $l$ and 1 and the constant term, we obtain:

$\begin{array}{l}{\lambda }_2{\text{e}}^{-r_{0}t}\left[P_t^1+\left(r_0-{\delta }^2-{\lambda }_2{P}^1\left(t\right)\right)P^1\left(t\right)\right]x^2\\ +\left[U_t^1+\left(2\alpha +{\eta }^2\right)U^1\left(t\right)-\left(P^1\left(t\right)+\frac{{\left(\delta +\rho \eta \right)}^2}{{\lambda }_2}\right)\frac{{\text{e}}^{r_{0}t}{\left(R^1\left(t\right)\right)}^2}{4P^1\left(t\right)}\right]l^2\\ +\left[R_t^1+\left(r_0+\alpha -{\delta }^2-\delta \rho \eta -{\lambda }_2{P}^1\left(t\right)\right)R^1\left(t\right)\right]xl\\ +{\lambda }_2{\text{e}}^{-r_{0}t}[P_t^1{Q}^1\left(t\right)+P^1\left(t\right)Q_t^1-(\left({\delta }^2+{\lambda }_2{P}^1\left(t\right)\right)Q^1\left(t\right)\\ -2\left(\left(1-\tilde{c}\right)C_1\left(t\right){\text{e}}^{\alpha t}-B^{\text{*}}{\text{e}}^{\beta t}\right)+{\lambda }_1)P^1\left(t\right)]x\\ +[V_t^1+\alpha V^1\left(t\right)+(\left(1-\tilde{c}\right)C_1\left(t\right){\text{e}}^{\alpha t}-B^{\text{*}}{\text{e}}^{\beta t}-\frac{{\lambda }_1}{2}\\ -\frac{1}{2}\left({\delta }^2+\delta \rho \eta +{\lambda }_2{P}^1\left(t\right)\right)Q^1\left(t\right))R^1\left(t\right)]l\\ +K_t^1+{\lambda }_2{\text{e}}^{-r_{0}t}{P}^1\left(t\right)Q^1\left(t\right)[\left(1-\tilde{c}\right)C_1\left(t\right){\text{e}}^{\alpha t}-B^{\text{*}}{\text{e}}^{\beta t}-\frac{{\lambda }_1}{2}\\ -\frac{1}{4}\left({\delta }^2+{\lambda }_2{P}^1\left(t\right)\right)Q^1\left(t\right)]-\frac{{\lambda }_1^2{\text{e}}^{-r_{0}t}}{4}=0\end{array}$

The coefficients of $x^2$, $l^2$, $xl$, $x$, $l$ and 1 and the constant term must vanish, which yields the following system of differential equations:

$P_t^1+\left(r_0-{\delta }^2-{\lambda }_2{P}^1\left(t\right)\right)P^1\left(t\right)=0$(34)

$U_t^1+\left(2\alpha +{\eta }^2\right)U^1\left(t\right)-\left(P^1\left(t\right)+\frac{{\left(\delta +\rho \eta \right)}^2}{{\lambda }_2}\right)\frac{{\text{e}}^{r_{0}t}{\left(R^1\left(t\right)\right)}^2}{4P^1\left(t\right)}=0$(35)

$R_t^1+\left(r_0+\alpha -{\delta }^2-\delta \rho \eta -{\lambda }_2{P}^1\left(t\right)\right)R^1\left(t\right)=0$(36)

$\frac{P_t^1}{P^1\left(t\right)}{Q}^1\left(t\right)+Q_t^1-\left({\delta }^2+{\lambda }_2{P}^1\left(t\right)\right)Q^1\left(t\right)-2\left(\left(1-\tilde{c}\right)C_1\left(t\right){\text{e}}^{\alpha t}-B^{\text{*}}{\text{e}}^{\beta t}\right)+{\lambda }_1=0$ (37)

$\begin{array}{l}{V}_t^1+\alpha V^1\left(t\right)+(\left(1-\tilde{c}\right)C_1\left(t\right){\text{e}}^{\alpha t}-B^{\text{*}}{\text{e}}^{\beta t}-\frac{{\lambda }_1}{2}\\ -\frac{1}{2}\left({\delta }^2+\delta \rho \eta +{\lambda }_2{P}^1\left(t\right)\right)Q^1\left(t\right))R^1\left(t\right)=0\end{array}$(38)

$\begin{array}{l}{K}_t^1+{\lambda }_2{\text{e}}^{-r_{0}t}{P}^1\left(t\right)Q^1\left(t\right)[\left(1-\tilde{c}\right)C_1\left(t\right){\text{e}}^{\alpha t}-B^{\text{*}}{\text{e}}^{\beta t}-\frac{{\lambda }_1}{2}\\ -\frac{1}{4}\left({\delta }^2+{\lambda }_2{P}^1\left(t\right)\right)Q^1\left(t\right)]-\frac{{\lambda }_1^2{\text{e}}^{-r_{0}t}}{4}=0\end{array}$(39)

with boundary conditions (32).

Differential Equations (34)-(36) with boundary conditions in (32) can be easily solved. First, because $R^1\left(T\right)=0$, solving (36) gives $R^1\left(t\right)=0$ for $0<t<T$. Consequently, we obtain from Equations (35) and (38) that $U^1\left(t\right)=V^1\left(t\right)=0$ for $0<t<T$. Second, solving (34) gives

$P^1\left(t\right)=\begin{array}{l}\{\begin{array}{ll}\frac{1}{{\lambda }_2\left(T-t\right)+1},\hfill & r_0={\delta }^2,\hfill \\ \frac{r_0-{\delta }^2}{{\lambda }_2+\left(r_0-{\delta }^2-{\lambda }_2\right){\text{e}}^{-\left(r-{\sigma }^2\right)\left(T-t\right)}},\hfill & r_0\ne {\delta }^2,\hfill \end{array}\hfill \end{array}$

Using (34) in (37), we obtain that $Q^1\left(t\right)$ satisfies

$Q^1\left(t\right)=\begin{array}{l}\{\begin{array}{ll}2{\text{e}}^{r_{0}t}\left[{\displaystyle {\int }_t^T\left(1-\tilde{c}\right)C_1\left(s\right){\text{e}}^{\left(\alpha -r_0\right)s}\text{d}s}-B^{*}\left(T-t\right)-x_0\right]\hfill & \hfill \\ +\frac{{\lambda }_1}{r_0}\left({\text{e}}^{-r_0\left(T-t\right)}-1\right),\hfill & r_0=\beta ,\hfill \\ 2{\text{e}}^{r_{0}t}[{\displaystyle {\int }_t^T\left(1-\tilde{c}\right)C_1\left(s\right){\text{e}}^{\left(\alpha -r_0\right)s}\text{d}s}-x_0\stackrel{}{}\hfill & \hfill \\ -\frac{B^{*}}{\beta -r_0}\left({\text{e}}^{\left(\beta -r_0\right)T}-{\text{e}}^{\left(\beta -r_0\right)t}\right)]+\frac{{\lambda }_1}{r_0}\left({\text{e}}^{-r_0\left(T-t\right)}-1\right),\hfill & r_0\ne \beta ,\hfill \end{array}\hfill \end{array}$

Finally, solving Equation (39) yields an explicit expression for $K^1\left(t\right)$:

$\begin{array}{l}{K}^1\left(t\right)=x_0^2{\text{e}}^{2r_{0}T}+{\displaystyle {\int }_t^T{\text{e}}^{-r_{0}s}}\{{\lambda }_2{P}^1\left(s\right)Q^1\left(s\right)[\left(1-\tilde{c}\right)C_1\left(s\right){\text{e}}^{\alpha s}\stackrel{}{}\\ -B^{\text{*}}{\text{e}}^{\beta s}-\frac{{\lambda }_1}{2}-\frac{1}{4}\left({\delta }^2+{\lambda }_2{P}^1\left(s\right)\right)Q^1\left(s\right)]-\frac{{\lambda }_1^2}{4}\}\text{d}s.\end{array}$

We now check the constraint given in (30), which is:

${\varphi }_{xx}^1=2{\lambda }_2{\text{e}}^{-r_{0}t}{P}^1\left(t\right)>0.$

Here $P^1\left(t\right)$ is defined by Equation (24), and ${\lambda }_2>0$. Clearly $P^1\left(t\right)>0$ when $r_0={\sigma }^2$. If $r_0\ne {\sigma }^2$, one can likewise verify $P^1\left(t\right)>0$ by treating the two cases $r_0>{\sigma }^2$ and $r_0<{\sigma }^2$ separately. □

Remark 3.1 It is notable that the optimal asset-allocation strategy ${\pi }_1^{{\psi }_2}\left(t,x,l\right)$ in (20) does not depend on the parameter ${\lambda }_2$, which measures aversion to intergenerational risk sharing.

Remark 3.2 It is also notable that the optimal asset-allocation strategy ${\pi }_1^{{\psi }_2}\left(t,x,l\right)$ given in (20) does not depend on the salary level $l$ at time $t$. From the optimal benefit-adjustment rule $f^{{\psi }_2}\left(t,x,l\right)$ in (21), one obtains the optimal aggregate benefit payment rate for all retirees at time $t$, denoted $B^{*}\left(t\right)$. This adjusted aggregate payment takes the following form ((4)):

$B^{\text{*}}\left(t\right)=I\left(t\right)f^{{\psi }_2}\left(t\right)l=\frac{1}{1-q\left(t\right)}\left[B^{\text{*}}{\text{e}}^{\beta t}+\frac{{\lambda }_1}{2}+{\lambda }_2{P}^1\left(t\right)\left(x+\frac{Q^1\left(t\right)}{2}\right)\right].$(40)

The expression above does not depend on the salary $l$ at time $t$. Likewise, the value function in (22) is independent of $l$, which implies that fluctuations in salary are fully offset by the optimal benefit-adjustment factor $f^{{\psi }_2}\left(t,x,l\right)$.

Remark 3.3 It is clear that the strategies ${\pi }_1^{{\psi }_2}\left(t,x,l\right)$, $f^{{\psi }_2}\left(t,x,l\right)$ and the value function ${\varphi }^1\left(t,x,l\right)$ depend on the policy ${\psi }_2=\left({\pi }_2(\cdot ),q(\cdot )\right)$ via the expressions in (21) and (23).

In this section, we solve the optimal control problem (15) and derive expressions of the leader’s optimal strategy ${\psi }_2^{\text{*}}$ within all the admissible policies ${\Psi }_2$.

The value function of reinsurer is defined as

$\varphi \left(t,y,l\right):=\underset{\left({\psi }_1^{*},{\psi }_2\right)\in \Psi }{\min }{J}_2\left(t,y,l;{\psi }_1^{*},{\psi }_2\right),t,y,l>0$(41)

where $J_2\left(t,y,l\right)$ is given by (12) and ${\psi }_1^{\text{*}}=\left({\pi }_1^{{\psi }_2}(\cdot ),f^{{\psi }_2}(\cdot )\right)$ is given by (20) and (21).

Same as the Section 3.1, we derive an HJB equation associated with the stochastic control problem (41) and get the following HJB equation satisfied by the value function $\varphi \left(t,x,l\right)$:

$\begin{array}{l}\underset{{\psi }_2\in {\Psi }_2}{\min }\{{\varphi }_t+\left[r_{0}y+\left(\mu -r_0\right){\pi }_2+\tilde{c}{C}_1\left(t\right){\text{e}}^{\alpha t}-ql{f}^{{\psi }_2}\left(t,y,l\right)I\left(t\right)\right]{\varphi }_y+\alpha l{\varphi }_l+\frac{1}{2}{\pi }^2{\sigma }^2{\varphi }_{yy}\begin{array}{c}\stackrel{}{}\\ \end{array}\\ +\frac{1}{2}{\eta }^2{l}^2{\varphi }_{ll}+\rho \sigma \eta l\pi {\varphi }_{yl}^2+\left[{\left(\frac{q\left(t\right)}{1-q\left(t\right)}-\theta \left(t\right)\right)}^2-{\lambda }_3\left(\frac{q\left(t\right)}{1-q\left(t\right)}-\theta \left(t\right)\right)\right]\}=0\end{array}$ (42)